Undecidable problem

About: Undecidable problem is a research topic. Over the lifetime, 3135 publications have been published within this topic receiving 71238 citations.

On the Equivalence of Two Systems of Affine Recurrence Equations

Abstract: This paper deals with the problem of deciding whether two Systems of Affine Recurrence Equations are equivalent or not. A solution to this problem would be a step toward algorithm recognition, an important tool in program analysis, optimization and parallelization. We first prove that in the general case, the problem is undecidable. We then show that there nevertheless exists a semi-decision procedure, in which the key ingredient is the computation of transitive closures of affine relations. This is a non-effective process which has been extensively studied. Many partial solutions are known. We then report on a pilot implementation of the algorithm, describe its limitations, and point to unsolved problems.

Reasoning about strings in databases

Abstract: In order to enable the database programmer to reason about relations over strings of arbitrary length we introduce alignment logic, a modal extension of relational calculus. In addition to relations, a state in the model consists of a two-dimensional array where the strings are aligned on top of each other. The basic modality in the language (a transpose, or “slide”) allows for a rearrangement of the alignment, and more complex formulas can be formed using a syntax reminiscent of regular expressions, in addition to the usual connectives and quantifiers. It turns out that the computational counterpart of the string-based portion of the logic is the class of multitape two-way finite state automata, which are devices particularly well suited for the implementation of string matching. A computational counterpart of the full logic is obtained from relational algebra by extending the selection operator into filters based on these multitape machines. Safety of formulas in alignment logic implies that new strings generated from old ones have to be of bounded length. While an undecidable property in general, this boundedness is decidable for an important subclass of formulas. As far as expressive power is concerned, alignment logic includes previous proposals for querying string databases, and gives full Turing computability. The language can be restricted to define exactly regular sets and sets in the polynomial hierarchy.

Multi-Dimensional Description Logics

Abstract: In this paper, we construct a new concept description language intended for representing dynamic and intensional knowledge The most important feature distinguishing this language from its predecessors in the literature is that it allows applications of modal operators to all kinds of syntactic terms: concepts, roles and formulas Moreover, the language may contain both local (ie, state-dependent) and global (ie, state-independent) concepts, roles and objects All this provides us with the most complete and natural means for reflecting the dynamic and intensional behaviour of application domains We construct a satisfiability checking (mosaic-type) algorithm for this language (based on ALC) in (i) arbitrary multimodal frames, (ii) frames with universal accessibility relations (for knowledge) and (iii) frames with transitive, symmetrical and euclidean relations (for beliefs) On the other hand, it is shown that the satisfaction problem becomes undecidable if the underlying frames are arbitrary linear orders or the language contains the common knowledge operator for n ≥ 2 agents

Inclusion is Undecidable for Pattern Languages

Abstract: The inclusion problem for (nonerasing) pattern languages was raised by Angluin [1] in 1980. It has been open ever since. In this paper, we settle this open problem and show that inclusion is undecidable for (both erasing and nonerasing) pattern languages. In addition, we show that a special case of the inclusion problem, i.e., the inclusion problem for terminal-free erasing pattern languages, is decidable.

Specifying role interaction in concept languages

Abstract: The KL-ONE concept language provides role-value maps (RVMs) as a concept forming operator that compares sets of role fillers. This is a useful means to specify structural properties of concepts. Recently, it has been shown that concept languages providing RVMs together with some other common concept-forming operators induce an undecidable subsumption problem. Thus, RVMs have been restricted to chainings of functional roles as, for example, in CLASSIC. Although this restricted RVM is still a useful operator, one would like to have additional means to specify interaction of general roles. The present paper investigates two concept languages for that purpose. The first one provides concept forming operators that generalize the restricted RVM in a different direction. Unfortunately, it turns out that this language also has an undecidable subsumption problem. The second formalism allows to specify structural properties w.r.t. roles without using general equality and is equipped with (complete) decision procedures for its associated reasoning problems.